Newton Raphson

Any help with this would be greatly appreciated... I haven't been using maths since I finished high school but have a maths test to do tomorrow as part of an interview. Here is a sample question, and I have no idea where to begin with it!

Provide reasoning that the equation:

10 = exp(-2x) + 2^x

has at least one solution in the range 0 <x < 5

Use the Newton Raphson technique to find an approximate solution to the equation.

The method is quite easy actually, it can just be tedious because you have to do many iterations of the calculation to get a good approximation. The basic idea is that you make an initial guess of the zero, which is x_0 , or x_1 from your post. Then you use that guess to find a closer guess, x_2. Now use x_2 as your guess and you'll get x_3, etc. etc. It can take 20-30 iterations for some functions and 10ish for others, really depends on how close you want to get.

Any help with this would be greatly appreciated... I haven't been using maths since I finished high school but have a maths test to do tomorrow as part of an interview. Here is a sample question, and I have no idea where to begin with it!

Provide reasoning that the equation:

10 = exp(-2x) + 2^x

has at least one solution in the range 0 <x < 5

This part, at least, is easy. If x= 0, exp(-2(0))+ 2^0= 1+ 1= 2. If x= 5, exp(-2(5))+ 2^5= exp(-10)+ 32= 32+ some positive number. Since f(x)= exp(-2x)+ 2^x is the sum of continuous functions, it is continuous itself so, since 10 is between 2 and 32, there must be some x, between 0 and 5 so that exp(-2x)+ 2^x= 10.

Use the Newton Raphson technique to find an approximate solution to the equation.

Any help with this would be greatly appreciated... I haven't been using maths since I finished high school but have a maths test to do tomorrow as part of an interview. Here is a sample question, and I have no idea where to begin with it!

Out of curiosity, what kind of job is it? If you don't mind my asking, that is.